What do you mean by saves to your browser storage?
On computer, mobile, Or both?
Sorry you're having issues saving the game! When you save from the main menu screen that data should save to the cache of whatever browser you’re using - I'd recommend downloading onto desktop and opening in Chrome for the most reliable results (I know Twine doesn't work great in some browsers such as Safari)
No worries! To be honest I'm not an expert on the technical details myself, but I think it's a bit like how your search history is saved by whatever browser you're using (like Chrome, Safari, etc.)? I'd recommend downloading the .html file and opening it directly instead of opening through the itch.io page. Hope that helps!